Adobe Illustrator does not make this process obvious, and the built-in measure tool can measure only one line segment at a time.
To find the lengths of a group of lines (grouped or selected), turn on the Document Info dialog using the Window menu.
Select the options drop-down menu (the icon looks like justified text) on the upper right-hand corner of the Document Info dialog. Select Objects.
Now you'll have the nitty-gritty details about your selected path(s)!
This could also be useful for measuring distances on non-georeferenced raster maps (with an absolute scale) by using a series of line paths.